I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection $$X' \to X$$ with the following two additional properties:
- $X'$ is quasi-compact and Hausdorff;
- The cardinality of $X'$ is not too much bigger than that of $X$, say something like $|X'| < 2^{2^{2^{|X|}}}$ would suffice;
- the association $X \mapsto (X' \to X)$ is "weakly functorial" in the sense that once we are given a map $S \to X$ between quasi-compact $T_1$ topological spaces, we may produce a commutative diagram:$$\require{AMScd}\begin{CD} S' @>>> X' \\ @VVV @VVV \\ S @>>> X \end{CD}$$
Note that the Stone-Cech compactification doesn't work since $X$ is not Hausdorff, and Hausdorf quotient of $X$ is going the wrong direction.
The motivation for this is to give an alternative proof of Clausen-Scholze's lecture notes on Condensed Mathematics Proposition 2.15 (first half): The statement there, roughly speaking, is that given any quasi-compact $T_1$ topological space $X$ which satisfies some cardinality bound $|X| < \kappa$ for certain kind of strong limit cardinal $\kappa$, and suppose we are given some map $f: S \to X$ for some extremally disconnected set $S$, then one can factorize $f$ as $S \to X' \to X$ where $X'$ is a quasi-compact Hausdorff space with $|X'| < \kappa$ as well.
If the above "weakly functorial resolution" exists, then one can prove the said Proposition rather easily. (The current proof is indeed very short, except I can't quite understand it...)
